Cohomology of bundles on homological Hopf manifolds (Q983639)

The author considers the notion of \(( \mathbb Q) \) homological Hopf manifold and proves the following result: Let \(\mathcal K\) be a homological Hopf manifold with the universal covering space \(p: V \rightarrow \mathcal K\) (here \(V\) is a complement of a compact set \(A\) in a Stein space \(\tilde{V}\) and the inclusion is denoted by \( i:V \rightarrow \tilde{V}\)). Let \(f\) be an element of infinite order in \( \pi_{1} (\mathcal K)\) and denote by \(f_{V}\) the corresponding biholomorphic automorphism of \(V\). If \(E\) is a bundle on \(\mathcal K\), denote by \( f^{k}_{V}\) the corresponding automorphism of \( H^{k}(V, p^{\star}(E))\) and assume that the local cohomology group \(H^{r}_{A} ( \tilde{V},i_{\star}(\pi ^{\star} (E)))\) is finite dimensional. Then dim \(H^{k}(\mathcal K, E) = \) dim Ker \(f^{k}_{V} + \) dim Coker \(f^{k-1}_{V}\). In particular, if \( H^{k}(V, p^{\star}(E))=0\) for \(a \leq k \leq b\) then \(H^{k}(\mathcal K, E)=0\) for \(a+1 \leq k \leq b-1\). Thus, informations concerning properties of cohomology of bundles on generalizations of Hopf manifolds are obtained. For example, for certain homological Hopf manifolds \(\mathcal K\), Pic(\(\mathcal K\)) =\( \mathbb C ^{\star}\). Several types of discrete invariants on generalized Hopf manifolds are presented. The author considers also the degeneration of Hodge-de Rham spectral sequence for these manifolds. Many examples are considered throughout this interesting paper.